The Law of the Euler Scheme for Stochastic Differential Equations: II. Convergence Rate of the Density

In the first part of this work [4] we have studied the approximation problem of Ε/(Χχ) by Ef(X%), where (Xt) is the solution of a stochastic differential equation, (X?) is defined by the Euler discretization scheme with step ^, and /(·) is a given function, only supposed measurable and bounded; we have proven that the error can be expanded in terms of powers of £, under a nondegeneracy condition of Hormander type for the infinitesimal generator of (Xt)· In this second part, we consider the density of the law of a small perturbation of Χγ and we compare it to the density of the law of XT: we prove that the difference between the densities can also be expanded in terms of ^. The results of this paper had been announced in special issues of journals devoted to the Proceedings of Conferences: see Bally, Protter and Talay [2] and Bally and Talay [3]. AMS(MOS) classification: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.